Character-Jump Heuristic: During the testing of a possible placement of the pattern within the text, a mismatch of character text[i]=c with the corresponding character pattern[k] is handled as follows. If c is not contained anywhere in the pattern, then shift the pattern completely past text[i] = c. Otherwise, shift the pattern until an occurrence of character c gets aligned with text[i].

when a mismatch is found near the right end of the pattern, we may end up realigning the pattern beyond the mismatch, without ever examining several characters of the text preceding the mismatch.

Notice that when the characters e and i mismatch at the right end of the original placement of the pattern, we slide the pattern beyond the mismatched character, without ever examining the first four characters of the text.

The original comparison results in a mismatch with character e of the text. Because that character is nowhere in the pattern, the entire pattern is shifted beyond its location. The second comparison is also a mismatch, but the mismatched character s occurs elsewhere in the pattern. The pattern is then shifted so that its last occurrence of s is aligned with the corresponding s in the text.

If the mismatched character occurs elsewhere in the pattern, we must consider two possible subcases depending on whether its last occurrence is before or after the character of the pattern that was mismatched.

we slide the pattern only one unit. It would be more productive to slide it rightward until finding another occurrence of mismatched character text[i] in the pattern, but we do not wish to take time to search for another occurrence. The efficiency of the Boyer-Moore algorithm relies on quickly determining where a mismatched character occurs elsewhere in the pattern. In particular, we define a function last(c) as
If c is in the pattern, last(c) is the index of the last (rightmost) occurrence of c in the pattern. Otherwise, we conventionally define last(c) = −1.

Additional rules for the character-jump heuristic of the Boyer-Moore algorithm. We let i represent the index of the mismatched character in the text, k represent the corresponding index in the pattern, and j represent the index of the last occurrence of text[i] within the pattern. We distinguish two cases: (a) j < k, in which case we shift the pattern by k − j units, and thus, index i advances by m − (j + 1) units; (b) j > k, in which case we shift the pattern by one unit, and index i advances by m − k units.

We prefer to use a hash table to represent the last function, with only those characters from the pattern occurring in the map. The space usage for this approach is proportional to the number of distinct alphabet symbols that occur in the pattern, and thus O(max(m, |Σ|)).

The correctness of the Boyer-Moore pattern-matching algorithm follows from the fact that each time the method makes a shift, it is guaranteed not to “skip” over any possible matches.

If using a traditional lookup table, the worst-case running time of the Boyer-Moore algorithm is O(nm + |Σ|). The computation of the last function takes O(m+|Σ|) time, although the dependence on |Σ| is removed if using a hash table. The actual search for the pattern takes O(nm) time in the worst case

The original algorithm achieves worst-case running time O(n + m + |Σ|) by using an alternative shift heuristic for a partially matched text string, whenever it shifts the pattern more than the character-jump heuristic. This alternative shift heuristic is based on applying the main idea from the Knuth-Morris-Pratt pattern-matching algorithm.